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N. A. Carella

Publications and source records attributed to N. A. Carella.

At least 19 recordsLinked to original sources
arXiv2026

Note on the Exceptional Set in the ABC Conjecture

Fix $\varepsilon>0$, let $x>1$ be a large real number and let $\text{rad}(n)=\prod_{p\mid n}p$ be the radical of an integer $n\geq1$. A triple $(a,b,c)$, with $a+b=c$ and $\gcd(a,b,c)=1$, such that $c>(\text{rad}(abc))^{1+\varepsilon}$, is called exceptional triple. Recent works have proved that the cardinality $\#\mathscr{E}(x)$ of set $\mathscr{E}$ of exceptional triples satisfies $\#\mathscr{E}(x)=O(x^{33/50})$. This note proves that the cardinality of the exceptional set $\mathscr{E}(x)$ of triples $(a,b,c)$ is an infinite set unconditionally for any $\varepsilon<2/35$.

math.GM↗
arXiv2026

Primitive Roots In Short Intervals

Let $p\geq 2$ be a large prime, and let $N\gg ( \log p)^{1+\varepsilon}$. This note proves the existence of primitive roots in the short interval $[M,M+N]$, where $M \geq 2$ is a fixed number, and $ \varepsilon>0$ is a small number. In particular, the least primitive root $g(p)= O\left ((\log p)^{1+\varepsilon} \right)$, and the least prime primitive root $g^*(p)= O\left ((\log p)^{1+\varepsilon} \right)$ unconditionally.

math.GM↗
arXiv2026

Least Consecutive Pair of Quadratic Nonresidues

Let $p>1$ be a large prime number and let $x=O((\log p)^2(\log\log p)^3$ be a real number. It is proved that the least consecutive pair of quadratic nonresidues $u\ne\pm1, v^2$ and $u+1$ satisfies the upper bound $u\ll x$ in the prime field $\mathbb{F}_p$.

math.GM↗
arXiv2026

Linear Independence Of Some Irrational Numbers

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of triples $\{1, e, π\}$, $\{1, e, π^{-1}\}$, and $\{1, π^r, π^s\}$, where $1\leq r<s $ are fixed integers.

math.GM↗
arXiv2026

Least Consecutive Pair of Primitive Roots

Let $p>1$ be a large prime number and let $x=O((\log p)^2(\log\log p)^5$ be a real number. It is proved that the least consecutive pair of primitive roots $u\ne\pm1, v^2$ and $u+1$ satisfies the upper bound $u\ll x$ in the prime field $\mathbb{F}_p$.

math.GM↗
arXiv2026

Simple Proof of the Primitive Root Conjecture

Let \(u\neq \pm 1,v^2\) be a fixed integer, let \(p\geq 2\) be a prime, and let $\text{ord}_p(u) \mid p-1$ be the multiplicative order of $u \text{ mod } p$. Define a prime counting function by $π(u,x)=\# \{ p\leq x:\text{ord}_p(u)=p-1 \}$. In 1967 Hooley proved a conditional asymptotic formula $π(u,x)=δ(u)x(\log x)^{-1}+O(\log\log x(\log x)^{-2}$ for the primitive root conjecture. This note proves an unconditional asymptotic formula $π(u,x)=δ(u)x(\log x)^{-1}+O(x(\log x)^{-2}$ of the same result, where $δ(u)>0$ is the density constant.

math.NT↗
arXiv2026

Equidistribution of Primitive Normal Elements in Finite Fields

Let $q=p^k$ be a prime power, let $n\geq2$ be an integer and let $\mathbb{F}_{q^n}$ be a finite field. It is shown that the set of primitive normal elements is a Salem set. Furthermore, it is proved that this set is strongly equidistributed in the finite field. Similar results are proved for the set of quadratic residues and the set of primitive roots modulo a large prime $p\geq 3$.

math.GM↗
arXiv2026

Small Primitive Normal Elements in Finite Fields

Let $q=p^k$ be a prime power, let $\mathbb{F}_q$ be a finite field and let $n\geq2$ be an integer. This note investigates the existence small primitive normal elements in finite field extensions $\mathbb{F}_{q^n}$. It is shown that a small nonstructured subset $\mathcal{A}\subset \mathbb{F}_{q^n}$ of cardinality $\#\mathcal{A}\gg (\log q^n) (\log\log q^n)^{1+\varepsilon}) $, where $\varepsilon>0$ is a small number, contains a primitive normal element.

math.GM↗
arXiv2025

Small Prime $k$th Power Residues and Nonresidues in Arithmetic Progressions

Let $p$ be a large odd prime, let $x=\log p)(\log\log p)^{3+\varepsilon}$ and let $q\ll\log\log p$ be an integer, where $\varepsilon>0$ is a small number. This note proves the existence of small prime quadratic residues and small prime quadratic nonresidues in the arithmetic progression $a+qm\ll x$, with relatively prime $1\leq a<q$, unconditionally. The same results are generalized to small prime $k$th power residues and nonresidues, where $k\mid p-1$ and $k\ll\log\log p$.

math.GM↗
arXiv2025

Upper Bound of the Least Quadratic Nonresidues

Let $p\geq3$ be a large prime and let $n(p)\geq2$ denotes the least quadratic nonresidue modulo $p$. This note sharpens the standard upper bound of the least quadratic nonresidue from the unconditional upper bound $n(p)\ll p^{1/4\sqrt{e}+\varepsilon}$ to the conjectured upper bound $n(p)\ll (\log p)^{1+\varepsilon}$, where $\varepsilon>0$ is a small number, unconditionally. This improvement breaks the exponential upper bound barrier.

math.GM↗
arXiv2025

Primitive Root Conjecture in Arithmetic Progressions

Let $x\geq 1$ be a large number, and let $1 \leq a 0$ constant. This note proves that the counting function for the number of primes $p \in \{p=qn+a: n \geq1 \}$ with a fixed primitive root $u\ne \pm 1, v^2$ has the asymptotic formula $π_u(x,q,a)=δ(u,q,a)x/ \log x +O(x/\log^b x),$ where $δ(u,q,a)>0$ is the density, and $b=b(c)>1$ is a constant.

math.GM↗
arXiv2025

Small Prime Primitive Roots in Arithmetic Progressions

Let $p>1$ be a large prime number, let $q=O(\log\log p)$ and let $1\leq a<q$ be a pair of relatively prime integers. It is proved that there is a prime primitive root $u\ll (\log p)(\log \log p)^5$ such that $u\equiv a\bmod q$ in the prime finite field $\mathbb{F}_p$.

math.GM↗
arXiv2025

Largest Prime Factors of Polynomials

Let $x>1$ be a large number. This note shows that the largest prime factor of the quadratic product $\prod_{x\leq n\leq 2x}\left(n^2+1 \right)$ satisfies the relation $p \geq x^{3/2}$ as $x$ tends to infinity. This improves the current estimate $p \geq x^{1.279}$.

math.GM↗
arXiv2025

Simultaneous Primitive Roots over Finite Rings

This note investigates the average density of prime numbers $p\in[x,2x]$ with respect to a random simultaneous primitive root $g\leq p^{1/2+\varepsilon}$ over the finite rings $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/p^2\mathbb{Z}$ as $x \to \infty$.

math.GM↗